Theorem univ 4461

Description: The union of the universe is the universe. Exercise 4.12(c) of [Mendelson] p. 235. (Contributed by NM, 14-Sep-2003.)

Assertion

Ref	Expression
univ	⊢ ∪ V = V

Proof of Theorem univ

Step	Hyp	Ref	Expression
1		pwv 3795	. . 3 ⊢ 𝒫 V = V
2	1	unieqi 3806	. 2 ⊢ ∪ 𝒫 V = ∪ V
3		unipw 4202	. 2 ⊢ ∪ 𝒫 V = V
4	2, 3	eqtr3i 2193	1 ⊢ ∪ V = V

Syntax hints:   = wceq 1348  Vcvv 2730  𝒫 cpw 3566  ∪ cuni 3796

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-uni 3797

This theorem is referenced by: (None)